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Derivatives Practice Tests

These Grade 12 Math practice tests focus on derivatives as instantaneous rates of change, including derivative rules, tangent lines, implicit differentiation, related rates, motion, extrema, concavity, optimization, and inverse functions.

Skills covered

  • Applying power, product, quotient, chain, trigonometric, exponential, and logarithmic derivative rules
  • Interpreting derivatives as slopes, velocity, acceleration, marginal quantities, and monotonicity
  • Using critical numbers, first and second derivative tests, concavity, and the Mean Value Theorem
  • Solving implicit differentiation and related-rates problems with correct units
  • Applying tangent-line approximation, inverse derivatives, parametric derivatives, and accumulation functions

How to use these practice tests

First identify the requested derivative or rate. Then choose the appropriate rule, simplify the expression, and interpret the sign and units in the original context.

For more Grade 12 Math practice, return to the Grade 12 Math category.

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Grade 12 Derivatives Practice Tests

Practice Grade 12 Math questions about derivative definitions, power, product, quotient and chain rules, trigonometric and logarithmic derivatives, implicit differentiation, related rates, extrema, concavity, and tangent lines.

Choose the best answer. Identify the derivative rule or interpretation, calculate carefully, and check signs, units, domains, and critical-point conditions.

10 Questions
0/10 Questions0 %
Correct 0
Incorrect 0
Unanswered 10
1. QuestionMultiple choice

What is the second derivative of f(x)=x4f(x)=x^4?

Please select an answer first.

Correct answer: A

The correct answer is 12x212x^2.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: f'(x)=4x^3 and f''(x)=12x^2.
  3. Interpret the result: The correct answer is 12x212x^2.

f'(x)=4x^3 and f''(x)=12x^2.

2. QuestionMultiple choice

If position is s(t)=t3s(t)=t^3, what is acceleration at t=2?

Please select an answer first.

Correct answer: B

The correct answer is 12.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Velocity is 3t^2 and acceleration is 6t, so a(2)=12.
  3. Interpret the result: The correct answer is 12.

Velocity is 3t^2 and acceleration is 6t, so a(2)=12.

3. QuestionMultiple choice

What is one Newton's method update for solving f(x)=0?

Please select an answer first.

Correct answer: C

The correct answer is xn+1=xn−f(xn)f′(xn)x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Newton's method uses the tangent-line x-intercept.
  3. Interpret the result: The correct answer is xn+1=xn−f(xn)f′(xn)x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}.

Newton's method uses the tangent-line x-intercept.

4. QuestionMultiple choice

What is ddx[ln⁡(f(x))]\frac{d}{dx}[\ln(f(x))]?

Please select an answer first.

Correct answer: D

The correct answer is f′(x)f(x)\frac{f'(x)}{f(x)}.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The chain rule gives f'(x)/f(x).
  3. Interpret the result: The correct answer is f′(x)f(x)\frac{f'(x)}{f(x)}.

The chain rule gives f'(x)/f(x).

5. QuestionMultiple choice

If f is one-to-one and differentiable, what is (f−1)′(a)(f^{-1})'(a)?

Please select an answer first.

Correct answer: A

The correct answer is 1f′(f−1(a))\frac{1}{f'(f^{-1}(a))}.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The derivative of an inverse is the reciprocal derivative evaluated at the corresponding input.
  3. Interpret the result: The correct answer is 1f′(f−1(a))\frac{1}{f'(f^{-1}(a))}.

The derivative of an inverse is the reciprocal derivative evaluated at the corresponding input.

6. QuestionMultiple choice

For parametric equations x=t^2 and y=t^3, what is dydx\frac{dy}{dx} when t=2?

Please select an answer first.

Correct answer: B

The correct answer is 3.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: dy/dx=(dy/dt)/(dx/dt)=3t^2/(2t)=3t/2, which is 3 at t=2.
  3. Interpret the result: The correct answer is 3.

dy/dx=(dy/dt)/(dx/dt)=3t^2/(2t)=3t/2, which is 3 at t=2.

7. QuestionMultiple choice

If f′(x)=2xf'(x)=2x, what is the rate of change of f at x=-3?

Please select an answer first.

Correct answer: C

The correct answer is -6.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Substitute x=-3 into f'(x)=2x.
  3. Interpret the result: The correct answer is -6.

Substitute x=-3 into f'(x)=2x.

8. QuestionMultiple choice

What is the normal-line slope when the tangent slope is 4?

Please select an answer first.

Correct answer: D

The correct answer is −14-\frac{1}{4}.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: Perpendicular nonvertical lines have slopes whose product is -1.
  3. Interpret the result: The correct answer is −14-\frac{1}{4}.

Perpendicular nonvertical lines have slopes whose product is -1.

9. QuestionMultiple choice

If A(x)=∫2x1+t2 dtA(x)=\int_2^x \sqrt{1+t^2}\,dt, what is A'(3)?

Please select an answer first.

Correct answer: A

The correct answer is 10\sqrt{10}.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: The Fundamental Theorem gives A'(x)=√(1+x^2), so A'(3)=√10.
  3. Interpret the result: The correct answer is 10\sqrt{10}.

The Fundamental Theorem gives A'(x)=√(1+x^2), so A'(3)=√10.

10. QuestionMultiple choice

What does a derivative of zero across an interval imply about the original function?

Please select an answer first.

Correct answer: B

The correct answer is It is constant on that interval.

  1. Identify the derivative task: Determine whether the question asks for a derivative rule, a tangent slope, a rate, an extremum, concavity, or a related rate.
  2. Apply the derivative method: No instantaneous change anywhere on the interval means the function is constant there.
  3. Interpret the result: The correct answer is It is constant on that interval.

No instantaneous change anywhere on the interval means the function is constant there.

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